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Title: A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations

Abstract

Herein, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the Kronecker product structure of the linear systems. The proposed algorithm efficiently approximates the solutions in low-rank tensor format. Using standard Krylov subspace methods for the data in tensor format is computationally prohibitive due to the rapid growth of tensor ranks during the iterations. To keep tensor ranks low over the entire iteration process, we devise a rank-reduction scheme that can be combined with the iterative algorithm. The proposed rank-reduction scheme identifies an important subspace in the stochastic domain and compresses tensors of high rank on-the-fly during the iterations. The proposed reduction scheme is a coarse-grid method in that the important subspace can be identified inexpensively in a coarse spatial grid setting. The efficiency of the present method is displayed by numerical experiments on benchmark problems.

Authors:
 [1];  [1]
+ Show Author Affiliations
  1. Univ. of Maryland, College Park, MD (United States)
Publication Date:
Research Org.:
Univ. of Maryland, College Park, MD (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)
OSTI Identifier:
1598333
Grant/Contract Number:  
SC0009301; DMS-1418754
Resource Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 39; Journal Issue: 5; Journal ID: ISSN 1064-8275
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; low-rank approximation; tensor format; stochastic Galerkin method; finite elements; GMRES; preconditioning; algebraic multigrid

Citation Formats

Lee, Kookjin, and Elman, Howard C. A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations. United States: N. p., 2017. Web. doi:10.1137/16M1075582.
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Lee, Kookjin, & Elman, Howard C. A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations. United States. https://doi.org/10.1137/16M1075582
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Lee, Kookjin, and Elman, Howard C. Thu . "A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations". United States. https://doi.org/10.1137/16M1075582. https://www.osti.gov/servlets/purl/1598333.
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@article{osti_1598333,
title = {A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations},
author = {Lee, Kookjin and Elman, Howard C.},
abstractNote = {Herein, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the Kronecker product structure of the linear systems. The proposed algorithm efficiently approximates the solutions in low-rank tensor format. Using standard Krylov subspace methods for the data in tensor format is computationally prohibitive due to the rapid growth of tensor ranks during the iterations. To keep tensor ranks low over the entire iteration process, we devise a rank-reduction scheme that can be combined with the iterative algorithm. The proposed rank-reduction scheme identifies an important subspace in the stochastic domain and compresses tensors of high rank on-the-fly during the iterations. The proposed reduction scheme is a coarse-grid method in that the important subspace can be identified inexpensively in a coarse spatial grid setting. The efficiency of the present method is displayed by numerical experiments on benchmark problems.},
doi = {10.1137/16M1075582},
journal = {SIAM Journal on Scientific Computing},
number = 5,
volume = 39,
place = {United States},
year = {2017},
month = {10}
}
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https://doi.org/10.1137/16M1075582
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Works referenced in this record:

On solving elliptic stochastic partial differential equations
journal, August 2002

  • Babuška, Ivo; Chatzipantelidis, Panagiotis
  • Computer Methods in Applied Mechanics and Engineering, Vol. 191, Issue 37-38
  • DOI: 10.1016/S0045-7825(02)00354-7

Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations
journal, January 2004

  • Babuska, Ivo; Tempone, Raúl; Zouraris, Georgios E.
  • SIAM Journal on Numerical Analysis, Vol. 42, Issue 2
  • DOI: 10.1137/S0036142902418680

Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
journal, July 2016

  • Bachmayr, Markus; Cohen, Albert
  • Mathematics of Computation, Vol. 86, Issue 304
  • DOI: 10.1090/mcom/3132

A projection method to solve linear systems in tensor format: A PROJECTION METHOD TO SOLVE LINEAR SYSTEMS IN TENSOR FORMAT
journal, January 2012

  • Ballani, Jonas; Grasedyck, Lars
  • Numerical Linear Algebra with Applications, Vol. 20, Issue 1
  • DOI: 10.1002/nla.1818

Low-Rank Solution of Unsteady Diffusion Equations with Stochastic Coefficients
journal, January 2015

  • Benner, Peter; Onwunta, Akwum; Stoll, Martin
  • SIAM/ASA Journal on Uncertainty Quantification, Vol. 3, Issue 1
  • DOI: 10.1137/130937251

Algorithms for Numerical Analysis in High Dimensions
journal, January 2005

  • Beylkin, Gregory; Mohlenkamp, Martin J.
  • SIAM Journal on Scientific Computing, Vol. 26, Issue 6
  • DOI: 10.1137/040604959

Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations
journal, September 1982

  • Brooks, Alexander N.; Hughes, Thomas J. R.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 32, Issue 1-3
  • DOI: 10.1016/0045-7825(82)90071-8

Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition
journal, September 1970

  • Carroll, J. Douglas; Chang, Jih-Jie
  • Psychometrika, Vol. 35, Issue 3
  • DOI: 10.1007/BF02310791

Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces
journal, January 2014

  • Constantine, Paul G.; Dow, Eric; Wang, Qiqi
  • SIAM Journal on Scientific Computing, Vol. 36, Issue 4
  • DOI: 10.1137/130916138

Stochastic model reduction for chaos representations
journal, August 2007

  • Doostan, Alireza; Ghanem, Roger G.; Red-Horse, John
  • Computer Methods in Applied Mechanics and Engineering, Vol. 196, Issue 37-40
  • DOI: 10.1016/j.cma.2006.10.047

Stochastic Galerkin Matrices
journal, January 2010

  • Ernst, Oliver G.; Ullmann, Elisabeth
  • SIAM Journal on Matrix Analysis and Applications, Vol. 31, Issue 4
  • DOI: 10.1137/080742282

Krylov Subspace Methods for Linear Systems with Tensor Product Structure
journal, January 2010

  • Kressner, Daniel; Tobler, Christine
  • SIAM Journal on Matrix Analysis and Applications, Vol. 31, Issue 4
  • DOI: 10.1137/090756843

Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems
journal, October 2011

  • Kressner, Daniel; Tobler, Christine
  • SIAM Journal on Matrix Analysis and Applications, Vol. 32, Issue 4
  • DOI: 10.1137/100799010

Solving stochastic systems with low-rank tensor compression
journal, May 2012

Generalized spectral decomposition for stochastic nonlinear problems
journal, January 2009

Block-diagonal preconditioning for spectral stochastic finite-element systems
journal, April 2008

  • Powell, C. E.; Elman, H. C.
  • IMA Journal of Numerical Analysis, Vol. 29, Issue 2
  • DOI: 10.1093/imanum/drn014

An Efficient Reduced Basis Solver for Stochastic Galerkin Matrix Equations
journal, January 2017

  • Powell, C. E.; Silvester, D.; Simoncini, V.
  • SIAM Journal on Scientific Computing, Vol. 39, Issue 1
  • DOI: 10.1137/15M1032399

GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
journal, July 1986

  • Saad, Youcef; Schultz, Martin H.
  • SIAM Journal on Scientific and Statistical Computing, Vol. 7, Issue 3
  • DOI: 10.1137/0907058

Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs
journal, April 2011

Model Reduction Based on Proper Generalized Decomposition for the Stochastic Steady Incompressible Navier--Stokes Equations
journal, January 2014

  • Tamellini, L.; Le Maître, O.; Nouy, A.
  • SIAM Journal on Scientific Computing, Vol. 36, Issue 3
  • DOI: 10.1137/120878999

The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
journal, January 2002

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